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The 5th GTSS

GEOMETRY-TOPOLOGY SUMMER SCHOOL

İstanbul Center for Mathematical Sciences
July 6-18, 2020






First Week

 

TIME              SPEAKER                  TITLE
Jul 6-12
8-10
Marisa Fernández*
Lectures on G2 Geometry
Jul 6-10
10-12
Shimpei Kobayashi Harmonic maps, constant mean curvature surfaces and integrable systems
Jul 6-12
10-12
Giovanni Bazzoni Locally Conformally Symplectic and Kähler Geometry
Jul 6-12
2:30-4
Vamsi Pritham Pingali The Calabi Conjectures
Jul 6-12
4-5:30
Spiro Karigiannis* Moduli Space of G_2 Manifolds
Jul 6-12
5:30-7
Subhojoy Gupta Hyperbolic Manifolds and Teichmüller Theory
Jul 6-12
10-12
Andreas Savas-Halilaj* Minimal Submanifolds, Mean Curvature Flow
and Isotopy Problems

Jul 6-12
8-10
Craig Van Coevering
Introduction to Sasakian Geometry


Second Week

 

TIME              SPEAKER                  TITLE
Jul 13-18
8-10
Miles Reid Topics in Algebraic Geometry
Jul 13-18
10-12
Yoshinori Gongyo Topics in Fano Varieties and Rational Connectedness
Jul 13-18
2:30-4
Dmitry Kerner*
Singular Algebraic Curves and Varieties*
Jul 13-18
4-5:30
Emre Sertöz* Topics in Moduli of Curves*
Jul 13-18
5:30-7
Susumu Tanabé Monodromy and Periods of Algebraic Varieties
Jul 13-18
10-12
Gerard van der Geer* Abelian Varieties
Jul 13-18
4-5:30
Mustafa Kalafat
Representation Theory of Lie Algebras (The Lie algebra of G2)
Jul 13-18
5:30-7
Özgür Kelekçi
Introduction to Einstein-Maxwell Manifolds
Jul 13-18
7-8:30
Buket Can Bahadır
Special Metrics in Complex Analysis

Scientific Commitee

 

Vicente Cortés University of Hamburg, Germany
İzzet Coşkun University of Illinois at Chicago, USA
Ljudmila Kamenova Stony Brook University, USA
Lei Ni University of California at San Diego, USA
Tommaso Pacini University of Torino, Italy
Gregory Sankaran University of Bath, UK
Misha Verbitsky IMPA, Brasil

Organizing Commitee

 

Craig van Coevering Bosphorus University
İlhan İkeda Bosphorus University
Mustafa Kalafat Nesin Mathematical Village







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Information

The 5th GTSS will be held at the IMBM, İstanbul Center for Mathematical Sciences
which is established in the main (South) Campus of Boğaziçi University (Bosphorus).
The first photo on top of this webpage demonstrates the view from the venue.
There will be about 15 mini-courses of introductory nature, related to the Geometry-Topology research subjects.
In the middle of the week there is an excursion on the Bosphorus. Do not forget to get your swimsuit with yourself.



Application

Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to the summer school.
Daily expenses including bed, breakfast, lunch, dinner is around 20€.
You may stay at cheap hostels around Taksim(Nightlife) square and take the subway to the Campus easily.
We are trying to arrange accomodation on Campus as well.

Please fill out the application form to attend to the summer school.

Airport: İstanbul Airport - IST is the closest one. Take a bus from the airport to the 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Alternatively you can use Sabiha Gökçen Airport - SAW and take a bus from there to 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Navigate the link above for more detailed arrival and venue information.

Visas: Check whether you need a visa beforehand.






Abstracts



The Calabi Conjectures

We shall state, prove, and study applications of some of Calabi’s Conjectures.

Topics to be covered are as follows: Definition of complex manifolds and holomorphic maps, Examples. Kahler metrics and examples. Holomorphic line bundles, the canonical bundle and the first Chern class. Statement of Calabi’s volume conjecture, the representability of the first Chern class conjecture, and the Kahler-Einstein conjecture. Examples and applications. If time permits, a gentle introduction to the ideas behind the proof (method of continuity, a priori estimates).

Textbook and/or References: -

Prerequisites:

Manifold theory including Riemannian metrics and the Levi-Civita connection, complex analysis, and functional analysis. Some exposure to elliptic PDE and complex geometry will be immensely helpful but is not necessary.

Harmonic maps, constant mean curvature surfaces and integrable systems

In this lecture, we give an overview of homogeneous spaces, harmonic maps, integrable systems, constant mean curvature surfaces and their relations. First, we see how harmonic maps from a surface into a homogeneous space can be understood as integrable systems and they are related to constant mean curvature surfaces in space forms via Gauss maps (Ruh-Vilms theorem). Finally, we give a construction method of harmonic maps (and constant mean curvature surfaces) via integrable systems (the so-called DPW method).

Daily description is as follows.

1. Overview

2. Manifolds and homogeneous spaces

3. Integrable systems and harmonic maps

4. Constant mean curvature surfaces, Gauss maps and Ruh-Vilms theorem

5. Construction of harmonic maps via integrable systems

6. Advanced topics

Textbook, References or/and course webpage:

1.Shoichi Fujimori, Shimpei Kobayashi, Wayne Rossman, Loop Group Methods for Constant Mean Curvature Surfaces, Rokko Lectures in Mathematics, 17 (2005), v+118 pp. arXiv:math/0602570

2. Josef F. Dorfmeister, Franz Pedit, Hongyou Wu, Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6 (1998), no. 4, 633–668.

Prerequisites: Linear Algebra, Calculus

Level (erase some): Graduate Advanced undergraduate

Hyperbolic Manifolds and Teichmüller Theory



Monodromy and Periods of Algebraic Varieties

We discuss various topics pertaining to the topology of complex algebraic varieties, relying on classical techniques in complex algebraic geometry and algebraic topology. We will begin by introducing the Milnor fibration and construct the monodromy action on the homology of the fibers of the Milnor fibration. We will discuss vanishing cycles in this concrete setting. We will then proceed to discuss applications of these concepts in classical algebraic geometry, such as the theory of Lefschetz pencils. In the second part of the week, we will focus on the cohomological aspects of our subject. The concept of the Gauss-Manin connection will be introduced. For motivational purposes, we will discuss the classical theory of periods, Picard-Fuchs equations and similar topics in the context of the Legendre family of elliptic curves. A basic review of local systems and vector bundles will be given. Time permitting, Hodge Structures and more advanced topics may be introduced. If the need arises, tools from basic differential topology and homological algebra may also be discussed.

Textbook or/and course webpage:

1. Singular Points of Complex Hypersurfaces, J. Milnor, 1968

2.Griffiths-Harris, Principles of Algebraic Geometry

3. Otto Forster, Lectures on Riemann Surfaces, 1981

4. A Scrapbook of Complex Curve Theory, H. Clemens, 1980

5. Complex Algebraic Geometry and Hodge Theory Claire Voisin, 2002

6. Pierre Deligne, SGA 7, vol II

Advanced Topics

5. Mixed Hodge Structures, C.Peters, J.H.M. Steenbrink, 2008

6. Sheaves in Topology A. Dimca, 2004

Also see the papers:

The Topology of Complex Projective Varieties after S. Lefschetz, K.Lamotke, 1979

P.A. Griffiths, Periods of integrals on algebraic Manifolds 1970

Prerequisites:

Complex Analysis, Algebraic Topology, Intro to Algebraic Curves (preferred)

Complex Potential Theory

The aim is to cover as much as the first five chapters of the textbook.

1. Harmonic Functions

2. Subharmonic Functions

3. Potential Theory

4. The Dirichlet Problem

5. Capacity

Level: Graduate, advanced undergraduate

Textbook or/and course webpage:

Potential Theory in the Complex Plane, T. Ransford

Prerequisites:

Complex Analysis

Special metrics in Sasakian geometry

This course will begin with an introduction to Sasakian geometry, a type of metric contact structure which is an odd dimensional analogue of a Kähler structure on a complex manifold. We will then consider the problem of finding a special Sasakian metrics, such as Einstein, and more generally constant scalar curvature and extremal metrics.

Textbook or/and course webpage:

1. Boyer, Charles P.; Galicki, Krzysztof. Sasakian geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2008. xii+613 pp.

2. Futaki, Akito; Ono, Hajime; Wang, Guofang Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differential Geom. 83 (2009), no. 3, 585–635.

3. Collins, Tristan C.; Székelyhidi, Gábor. K-semistability for irregular Sasakian manifolds. J. Differential Geom. 109 (2018), no. 1, 81–109.

Prerequisites:

A first year graduated course in differential geometry should be sufficient. -->

Representation Theory of the Lie Algebra of G_2

In this lecture series we give an introduction to the representation theory of the Lie algebra of G_2. We review Lie groups, Lie algebras, Cartan subalgebra, Root systems. Then, starting from concrete examples we will work on sl(2,C), sl(3,C), sl(4,C) and finally sl(n,C) and their representations. We also go into their geometric interpretations. Also spend time on sp(2n,C) and so(m,C) if time permits.

Textbook or/and course webpage:

1. Fulton, William; Harris, Joe - Representation theory. A first course.
Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. xvi+551 pp. ISBN: 0-387-97527-6
Lectures 11,12,13,14 and 22.

Prerequisites:

Lie groups and Lie algebras.

Minimal Submanifolds, Mean Curvature Flow and Isotopy Problems

Many fundamental results in geometry and topology have been established through the development of minimal submanifold theory and geometric flow techniques. In this mini course, I will start by discussing minimal submanifolds and scalar/vectorial maximum principles for elliptic and parabolic PDEs. Then, I will use these tools to prove Bernstein type theorems for graphical minimal submanifolds. Finally, I will focus on the mean curvature flow in high codimensions and will demonstrate how to use this powerful method to derive topological results for maps between Riemannian manifolds.

Textbook or/and course webpage:

1. K. Smoczyk, Mean curvature flow in higher codimension: introduction and survey. Springer Proceedings in Mathematics, Vol 7, 231-274 (2012). Text also available on arXiv 1104.3222.

2. Y.-L. Xin, Minimal submanifolds and related topics, Nankai Tracts in Mathematics, Vol. 16 (2018).

Prerequisites:

Differential Geometry, Riemannian Geometry.

Einstein-Maxwell Manifolds

An Einstein manifold is a (pseudo-)Riemannian manifold (M,g) (a spacetime) such that the Ricci tensor is proportional to the metric tensor. Einstein manifolds are the solutions of Einstein's field equations for pure gravity with cosmological constant Λ (Lambda). Einstein-Maxwell manifolds, on the other hand, satisfy Einstein-Maxwell equations consisting of gravity and electromagnetism. These manifolds are not only interesting for physics but also for pure geometry since they are related to many important topics of Riemannian geometry such as Riemannian submersions, homogeneous Riemannian spaces, Riemannian functionals and their critical points, Yang-Mills theory, holonomy groups etc. In these lectures we aim to provide basics of Einstein manifolds and some parts of their classifications. We will also deal with Einstein-Maxwell equations and study on some explicit examples. Textbook or References:

1. A. L. Besse, “Einstein Manifolds”, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10. Springer, Berlin (1987).

2. C. LeBrun, “The Einstein–Maxwell equations, Kähler metrics, and Hermitian geometry”, J. Geom. Phys. 91, 163–171 (2015).

Prerequisites:

Basic Differential Geometry (not a must but preferable)

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Contact: ilaydabariss@gmail.com, berkanuze@gmail.com



Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society